Harder to win big in the lottery

The ‘Big Wednesday’ lottery has moved from 6 balls out of 45 to 6 balls out of 50, which reduces even further the chance of getting the division 1 prize.  To win division 1, you need 6 balls correct out of 6, plus a correct coin toss.  There are 8,145,060 ways to choose 6 balls out of 45, and about twice as many ways, 15,890,700, to choose 6 balls out of 50.  Adding in the coin toss halves the chance of winning: the chance of winning per ‘line’ used to be 1 in 16,290,120 and is now 1 in 31,781,400.   For a minimum $4 ticket, which gives 4 ‘lines’, the chance of a division 1 prize was 1 in 4,072,530 and is now 1 in 7,945,350.

One back-of-the-envelope way to get roughly the correct impact of the change is to note that the chance of matching a given ball has gone down about 10%: from 1/45 to 1/50.  Multiplying 90% by itself six times says that the chance of winning is 53% of what it was, a very good approximation to the actual ratio, which is 51%.  The Dominion Post had the correct change, but the computations they report seem to have the effect of the coin toss backwards, so all their probabilities are overly optimistic by a factor of four.

Of course, the other way to look at it is that your chance of not winning division 1 with a $4 ticket has gone from 99.99998% to 99.99999%. Hardly seems worth mentioning.

Here’s how I did the computations (in R), for anyone who might be interested or who might disagree:

Number of combinations of six balls

> choose(45,6)
[1] 8145060
> choose(50,6)
[1] 15890700

Double the number of combinations for the coin toss
> choose(45,6)*2
[1] 16290120
> choose(50,6)*2
[1] 31781400

Divide by 4, for the 4 lines giving four chances
> choose(45,6)*2/4
[1] 4072530
> choose(50,6)*2/4
[1] 7945350

How much harder has it got: exactly
> (choose(45,6)*2/4)/(choose(50,6)*2/4)
[1] 0.5125677

Back-of-the envelope calculation
> 0.9^6
[1] 0.531441

Turn them into probabilities
> 1-1/(choose(45,6)*2/4)
[1] 0.9999998
> 1-1/(choose(50,6)*2/4)
[1] 0.9999999